Approximations of Lov´
asz extensions and
their induced interaction index
Jean-Luc Marichal
Applied Mathematics Unit, University of Luxembourg 162A, avenue de la Fa¨ıencerie, L-1511 Luxembourg, Luxembourg
Pierre Mathonet
Department of Mathematics, University of Li`ege Grande Traverse 12, B37, B-4000 Li`ege, Belgium
Abstract
The Lov´asz extension of a pseudo-Boolean function f : {0, 1}n → R is defined on
each simplex of the standard triangulation of [0, 1]nas the unique affine function ˆf :
[0, 1]n→ R that interpolates f at the n + 1 vertices of the simplex. Its degree is that
of the unique multilinear polynomial that expresses f . In this paper we investigate the least squares approximation problem of an arbitrary Lov´asz extension ˆf by
Lov´asz extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman [14] and then solved explicitly by Grabisch, Marichal, and Roubens [11], giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of ˆf and we present some of its properties. It turns out
that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche [10].
Key words: pseudo-Boolean function, Lov´asz extension, discrete Choquet integral,
least squares approximation, interaction index.
Email addresses: jean-luc.marichal[at]uni.lu (Jean-Luc Marichal),
1 Introduction and problem setting
Real valued set functions are extensively used in both cooperative game theory and multicriteria decision making (see for instance [11] and the references therein). Indeed, in its characteristic form, a cooperative game on a finite set of players N := {1, . . . , n}, is a set function v : 2N → R such that v(∅) = 0 which assigns to each coalition S of players a real number v(S) representing the worth of S. In multicriteria decision making, when N represents a set of criteria, a capacity (also called non-additive measure) on N is a monotone set function v : 2N → R such that v(∅) = 0 and v(N) = 1 which assigns to each subset S of criteria its weight v(S).
There is a natural identification of real valued set functions v : 2N → R with pseudo-Boolean functions f : {0, 1}n → R through the equality v(S) =
f (1S), where 1S denotes the characteristic vector of S in {0, 1}n (whose ith component is 1 if and only if i ∈ S). Moreover, it is known [15] that any pseudo-Boolean function f : {0, 1}n → R has a unique expression as a multilinear polynomial in n variables f (x) = X S⊆N a(S) Y i∈S xi, (1)
where the set function a : 2N → R is simply the M¨obius transform (see for instance [21]) of v, which can be calculated through
a(S) = X
T ⊆S
(−1)|S|−|T |v(T ). (2)
Let Sn denote the family of all permutations σ on N. The Lov´asz extension ˆ
f : [0, 1]n → R of any pseudo-Boolean function f is defined on each n-simplex
Sσ := {x ∈ [0, 1]n | xσ(1)6 · · · 6 xσ(n)} (σ ∈ Sn),
as the unique affine function which interpolates f at the n + 1 vertices of Sσ (see Lov´asz [17, §3] and Singer [24, §2]).
The concept of Lov´asz extension was initially used in combinatorial optimiza-tion [8,17,24] and then in cooperative game theory [2,11]. In this paper we are mostly concerned by the interpretation of this concept in decision mak-ing, where it was observed in [18] that the Lov´asz extension of a monotone pseudo-Boolean function is nothing but a discrete Choquet integral [4,5]. Re-call that discrete Choquet integrals were recently proposed in decision making as aggregation functions that generalize the weighted arithmetic means by the taking into consideration of the interaction phenomena among input variables; see for instance [12,19].
As pointed out in Grabisch et al. [11], the Lov´asz extension of any pseudo-Boolean function f can be expressed as the following min-polynomial
ˆ
f (x) = X
S⊆N
a(S) min
i∈S xi.
Its degree is that of the multilinear polynomial (1) that expresses f .
Hammer and Holzman [14] investigated the approximation of a pseudo-Boolean function f by a multilinear polynomial of (at most) a specified degree. More precisely, fixing k ∈ N, with k 6 n, they defined the best kth approximation of f as the multilinear polynomial fk : {0, 1}n→ R of degree at most k which
minimizes X
x∈{0,1}n
[f (x) − g(x)]2
among all multilinear polynomials g of degree at most k. They obtained an explicit form of the solution for k = 1 and k = 2. For a general k, the explicit form of the solution was obtained by Grabisch et al. [11, §7]. It is written as
fk(x) = X S⊆N |S|6k ak(S) Y i∈S xi, where ak(S) = a(S) + (−1)k+|S| X T ⊇S |T |>k ³ |T |−|S|−1 k−|S| ´ 2|T |−|S| a(T ). (3) Hammer and Holzman [14] also noticed that a1({i}) identifies with the classical
Banzhaf power index [3] related to the ith player. More generally, Grabisch et al. [11] noticed that, for any S ⊆ N, the coefficient a|S|(S) identifies with the Banzhaf interaction index [22] related to the coalition S.
In this paper we investigate the least squares approximation of a given Lov´asz extension ˆf by a min-polynomial of (at most) a specified degree. More
pre-cisely, fixing k ∈ N, with k 6 n, we give explicit formulas (see Section 2) for the best kth approximation of ˆf , that is, the min-polynomial ˆfk : [0, 1]n → R of degree at most k which minimizes
Z
[0,1]n
h
ˆ
f (x) − ˆg(x)i2 dx
among all min-polynomials ˆg of degree at most k.
It is important to emphasize that this least squares problem has a relevant application in multicriteria decision making. Indeed, consider a multicriteria decision making problem modelled by a Choquet integral and its underlying capacity on a set of n criteria. It is sometimes useful to approximate this Choquet integral (which is, actually, the Lov´asz extension of the capacity) by
simpler aggregation functions such as Lov´asz extensions of degree k (k < n). Indeed this subclass of functions includes the Choquet integrals constructed from k-order capacities introduced by Grabisch [9].
By analogy with the Banzhaf interaction index obtained from the approxi-mations of f , we show (see Section 3) that the approxiapproxi-mations of ˆf give rise
to a new interaction index having appealing properties (see for instance [7]). Furthermore, it turns out that its corresponding power index identifies with a power index introduced axiomatically by Grabisch and Labreuche [10]. In order to avoid a heavy notation, cardinalities of subsets S, T, . . . are denoted whenever possible by the corresponding lower case letters s, t, . . ., otherwise by the standard notation |S|, |T |, . . . Also, for any S ⊆ N we define the function minS : [0, 1]n → R as
minS(x) := min i∈S xi,
with the convention that min∅ ≡ 1. Finally, we denote by FN the family of set functions v : 2N → R.
2 Approximations of Lov´asz extensions
We first recall the setting of our approximation problem. For any k ∈ {0, . . . , n}, we denote by Vk the set of all min-polynomials ˆfk : [0, 1]n → R of degree at most k, i.e., of the form
ˆ fk(x) = X S⊆N |S|6k ak(S) min i∈S xi.
Definition 1 Let ˆf ∈ Vn and k ∈ {0, . . . , n}. The best kth approximation of ˆ
f is the min-polynomial ˆfk∈ Vk that minimizes
Z
[0,1]n
h
ˆ
f (x) − ˆg(x)i2 dx
among all min-polynomials ˆg of Vk.
In this section we first discuss existence and uniqueness of the best kth ap-proximation of any function ˆf ∈ Vn (see Proposition 3). Then we provide a closed-form formula for this approximation (see Theorem 9).
We begin with the following straightforward proposition:
Proposition 2 For any k ∈ {0, . . . , n}, the set Vk is a linear subspace of
functions on [0, 1]n. Moreover, the set
Bk:= {minS | S ⊆ N, s 6 k}, (4)
forms a basis of Vk and hence dim(Vk) =
Pk s=0 ³ n s ´ .
Consider the usual inner product h· , ·i of C0([0, 1]n) given by
hf1, f2i :=
Z
[0,1]nf1(x) f2(x) dx.
This clearly induces an inner product in Vn, which in turn allows to define a norm k ˆf k := h ˆf , ˆf i1/2 and then a distance d( ˆf
1, ˆf2) := k ˆf1− ˆf2k in Vn.
According to this terminology, the best kth approximation of ˆf ∈ Vn is the solution ˆfk of the following least squares problem
min{k ˆf − ˆgk2 | ˆg ∈ V
k},
which amounts to a classical orthogonal projection problem.
From now on, for any k ∈ {0, . . . , n}, we denote by Akthe orthogonal projector from Vn onto Vk.
Proposition 3 For any ˆf ∈ Vn and any k ∈ {0, . . . , n}, the best kth
approx-imation of ˆf exists and is uniquely given by ˆfk = Ak( ˆf ). In particular, it is
characterized as the unique element ˆfk of Vk such that ˆf − ˆfk is orthogonal to
Vk, i.e., to each element of the basis Bk defined in (4).
This projection problem can be simplified because of the following result, which immediately follows from the fact that the subspaces Vk are nested: Proposition 4 The operators Ak, k = 0, . . . , n, are such that
k 6 k0 ⇒ A
k(Ak0( ˆf )) = Ak( ˆf ). (5)
We thus observe that Ak( ˆf ) can be attained from ˆf by successively carrying out the projections An−1, An−2, . . . , Ak. We will therefore search for the rela-tion that links any two consecutive projecrela-tions. This relarela-tion will enable us to compute gradually Ak( ˆf ) from ˆf .
Remark 5 Fix k ∈ {0, . . . , n − 1} and assume that ˆfk+1 = Ak+1( ˆf ) is given.
By (5), Ak( ˆf ) is the orthogonal projection of Ak+1( ˆf ) onto Vk and by
Propo-sition 3, it is given as the unique solution ˆfk ∈ Vk of the following system
To solve this projection problem, we need the following two lemmas. The first one yields an explicit expression of the inner product hminS, minTi of any two elements of Bn. The second lemma, which makes use of the first one, yields an explicit expression of the projections onto Vk of the elements of Bk+1\ Bk. Lemma 6 For any S, T ⊆ N, there holds
Z [0,1]nminS(x) minT(x) dx = 1 |S ∪ T | + 2 µ 1 s + 1+ 1 t + 1 ¶ . (6)
Proof. See Appendix A. 2
Lemma 7 For any S ⊆ N, with s = k + 1, we have
Ak(minS) = X T S (−1)k+t ³ k+t+1 k+1 ´ ³ 2k+2 k+1 ´ minT.
Proof. See Appendix B. 2
Proposition 8 Let k ∈ {0, . . . , n − 1}. Given the coefficients ak+1(S) of
Ak+1( ˆf ), the coefficients ak(S) of Ak( ˆf ) are given by
ak(S) = ak+1(S) + (−1)k+s ³ k+s+1 k+1 ´ ³ 2k+2 k+1 ´ X T ⊇S t=k+1 ak+1(T ). (7)
Proof. Using formula (5) and then Lemma 7, we obtain
Ak( ˆf ) = Ak(Ak+1( ˆf )) = X S⊆N s6k+1 ak+1(S) Ak(minS) = X S⊆N s6k ak+1(S) minS+ X S⊆N s=k+1 ak+1(S) Ak(minS) = X S⊆N s6k ak+1(S) minS+ X S⊆N s=k+1 ak+1(S) X T S (−1)k+t ³ k+t+1 k+1 ´ ³ 2k+2 k+1 ´ minT,
which proves the result. 2
We are now ready to derive the explicit form of the coefficients of Ak( ˆf ) in terms of the coefficients of ˆf . It is worth comparing this formula with the
Theorem 9 Let k ∈ {0, . . . , n}. The coefficients of Ak( ˆf ) are given from those of ˆf by ak(S) = a(S) + (−1)k+s X T ⊇S t>k ³ k+s+1 k+1 ´³ t−s−1 k−s ´ ³ k+t+1 k+1 ´ a(T ). (8)
Proof. We only have to prove that the coefficients ak(S) given in (8) fulfill the recurrence relation (7).
Let k ∈ {0, . . . , n} and fix S ⊆ N with s 6 k. By substituting (8) into (7), we obtain, after removing the common term a(S),
(−1)k+s X R⊇S r>k ³ k+s+1 k+1 ´³ r−s−1 k−s ´ ³ k+r+1 k+1 ´ a(R) = (−1)k+s+1 X R⊇S r>k+1 ³ k+s+2 k+2 ´³ r−s−1 k−s+1 ´ ³ k+r+2 k+2 ´ a(R) + (−1)k+s ³ k+s+1 k+1 ´ ³ 2k+2 k+1 ´ X T ⊇S t=k+1 " a(T ) + X R⊇T r>k+1 ³ 2k+3 k+2 ´ ³ k+r+2 k+2 ´a(R) # .
Let us show that this equality holds. Dividing through by (−1)k+s and then removing the terms corresponding to a(R), with r = k + 1, the equality be-comes X R⊇S r>k+1 ³ k+s+1 k+1 ´³ r−s−1 k−s ´ ³ k+r+1 k+1 ´ a(R) = − X R⊇S r>k+1 ³ k+s+2 k+2 ´³ r−s−1 k−s+1 ´ ³ k+r+2 k+2 ´ a(R) + X R⊇S r>k+1 ³ k+s+1 k+1 ´³ 2k+3 k+2 ´³ r−s k−s+1 ´ ³ 2k+2 k+1 ´³ k+r+2 k+2 ´ a(R).
Fix R ⊇ S, with r > k + 1, and consider the coefficients of a(R) in the previous equality. By equating them, we obtain an identity which can be easily checked. 2
ˆ
f (x) = 3
10
³
x1+ x2+ x3+ min{x1, x2} + min{x1, x3} + min{x2, x3}
´
− 21
25 min{x1, x2, x3} + 1
25 min{x1, x2, x3, x4}.
The best constant approximation is given by
(A0f )(x) =ˆ
137 250,
the best linear approximation by
(A1f )(x) =ˆ 1 100 + 89 250(x1+ x2+ x3) + 1 125x4
and the best min-quadratic approximation by
(A2f )(x) = −ˆ 27 700 + 803 1750(x1 + x2+ x3) − 8 875 x4 − 19 175 ³
min{x1, x2} + min{x1, x3} + min{x2, x3}
´
+ 2 175
³
min{x1, x4} + min{x2, x4} + min{x3, x4}
´
.
Before closing this section, we show that every approximation preserves the symmetry properties of the functions. For instance, in the previous example, we observe that the function ˆf and all its approximations are symmetric in
the variables x1, x2, and x3.
Let us establish this result in a more general setting. Let X be a nonempty closed convex set in a finite-dimensional inner product space V . For any u ∈ V , the distance between u and X is achieved at a unique point in X. We denote this point by AX(u) and call it the projection of u onto X (see for instance [16, Chap. 3, §3.1]).
Lemma 11 Let I : V → V be an isometry such that I(X) = X. Then I and
AX commute, i.e., I ◦ AX = AX ◦ I.
Proof. We clearly have I[AX(u)] ∈ X for any u ∈ V . Furthermore, for any
v ∈ X, we have
kI(u) − I[AX(u)]k = ku − AX(u)k 6 ku − vk = kI(u) − I(v)k, which shows that I[AX(u)] is the projection of I(u) onto X. 2
Definition 12 For any σ in Sn, we define the linear operator Pσ of Vn as
Pσf (xˆ 1, . . . , xn) := ˆf (xσ(1), . . . , xσ(n)).
We say that σ ∈ Sn is a symmetry of ˆf if Pσ( ˆf ) = ˆf .
Proposition 13 For any σ ∈ Sn, the operator Pσ is an isometry of Vn such
that Pσ(Vk) = Vk. In particular, the operators Pσ and Ak commute and Ak
preserves the symmetries of its arguments.
Proof. It is clear that Pσ(Vk) = Vkand that Pσ is an isometry, i.e., it satisfies
hPσf , Pˆ σˆgi = h ˆf , ˆgi ( ˆf , ˆg ∈ Vn).
Therefore, by Lemma 11, Pσ and Ak commute. In particular, if σ ∈ Sn is a symmetry of ˆf , then
Pσ[Ak( ˆf )] = Ak[Pσ( ˆf )] = Ak( ˆf ) and hence σ is also a symmetry of Ak( ˆf ). 2
3 A new interaction index
In cooperative game theory the concept of power index (or value) was intro-duced in the pioneering work of Shapley [23]. Roughly speaking, a power index on N is a function φ : FN × N → R that assigns to every player i ∈ N in a game v ∈ FN his/her prospect φ(v, i) from playing the game. The Shapley
power index of a player i ∈ N in a game v ∈ FN is given by
φSh(v, i) := X T ⊆N \{i} 1 n à n − 1 t !−1 [v(T ∪ {i}) − v(T )].
Another frequently used power index is the Banzhaf power index [3,6] which, for a player i ∈ N in a game v ∈ FN, is defined by
φB(v, i) :=
X
T ⊆N \{i} 1
2n−1 [v(T ∪ {i}) − v(T )].
The concept of interaction index, which is an extension of that of power index, was recently introduced axiomatically to measure the interaction phenomena among players. An interaction index on N is essentially a function I : FN × 2N → R that assigns to every coalition S ⊆ N of players in a game v ∈ F
its interaction degree. Various interaction indices have been introduced thus far in the literature: the Shapley interaction index [9], the Banzhaf interaction
index [13,22], and the chaining interaction index [20], which all belong to
the class of cardinal-probabilistic interaction indices (see Definition 14 below) newly axiomatized in [7].
For instance, the Banzhaf interaction index on N, which extends the concept of Banzhaf power index on N, is the mapping IB : FN × 2N → R defined by
IB(v, S) :=
X
T ⊆N \S 1
2n−s ∆Sv(T ),
where ∆Sv(T ) is the S-derivative of v at T defined for any disjoint subsets
S, T ⊆ N by
∆Sv(T ) :=
X
R⊆S
(−1)s−rv(R ∪ T ). (9)
This index can be easily expressed in terms of the M¨obius transform (2) of v as IB(v, S) = X T ⊇S 1 2t−s a(T ).
It is noteworthy that, besides the axiomatic approach presented in [13,22], the Banzhaf interaction index can also be defined from the Hammer-Holzman approximation problem. Indeed, as pointed out in [14] and [11, §7], by con-sidering the leading coefficients in (3) of the best sth approximation, for all
s = 0, . . . , n, we immediately observe that IB(v, S) = as(S).
In this section we use the same approach to define a new interaction index from our approximation problem of Lov´asz extensions. In this sense this new index can be seen as an analog of the Banzhaf interaction index.
Before going on, we recall the concept of cardinal-probabilistic interaction index [7].
Definition 14 A cardinal-probabilistic interaction index on N is a mapping
Ip : FN× 2N → R such that, for any S ⊆ N, there is a family of nonnegative
real numbers {ps t(n)}t=0,...,n−s satisfying Pn−s t=0 pst(n) = 1, such that Ip(v, S) = X T ⊆N \S ps t(n) ∆Sv(T ). (10)
It has been proved [7, §4] that a mapping Ip : FN × 2N → R of the form (10) is a cardinal-probabilistic interaction index on N if and only if, for any integer
s ∈ {0, . . . , n}, there exists a uniquely determined cumulative distribution
function Fs on [0, 1] such that
ps t(n) = Z 1 0 x t(1 − x)n−s−tdF s(x). (11) Moreover, defining qs t := pst−s(t) = R1
0 xt−sdFs(x) for all integers s, t such that 0 6 s 6 t 6 n, we have Ip(v, S) = X T ⊇S qs ta(T ). (12)
The following lemma provides conditions on arbitrary coefficients qs
t so that a mapping of the form (12) can be a cardinal-probabilistic interaction index. Lemma 15 Consider an infinite sequence {qs
t}t>s>0. Then the following three
conditions are equivalent: (i) we have Pm i=0(−1)i ³ m i ´ qs
t+i> 0 for all m > 0 and all t > s > 0,
(ii) there is a unique cumulative distribution function Fs on [0, 1] such that, for
any integer t > s > 0, qst = Z 1 0 x t−sdF s(x),
(iii) for any N = {1, . . . , n}, the function Ip : FN × 2N → R, defined in (12) is
a cardinal-probabilistic interaction index.
Moreover, when these conditions are satisfied, then the mapping Ip : FN × 2N → R, defined in (12), is of the form (10) and the coefficients qs
t and pst(n)
are linked through qs
t = pst−s(t) and ps t(n) = n X i=s+t (−1)i−s−t à n − s − t i − s − t ! qs i.
Proof. (i) ⇔ (ii) Follows immediately from the Hausdorff’s moment problem (see for instance Akhiezer [1, Theorem 2.6.4]).
(ii) ⇒ (iii) Rewriting (12) in terms of v gives
Ip(v, S) = X J⊇S qs j X L⊆J (−1)j−lv(L).
Ip(v, S) = X T ⊆N \S " X J⊇S∪T (−1)j−s−tqs j # X R⊆S (−1)s−rv(R ∪ T ) = X T ⊆N \S pst(n)∆Sv(T ), where the coefficients ps
t(n) are given by ps t(n) = n X j=s+t (−1)j−s−t à n − s − t j − s − t ! qs j = Z 1 0 x t(1 − x)n−s−tdF s(x), which shows that Ip is a cardinal-probabilistic interaction index. (iii) ⇒ (ii) According to formula (12), we have
Ip(v, S) = X T ⊇S qs t a(T ) = X T ⊇S · Z 1 0 x t−sdF s(x) ¸ a(T ),
which completes the proof by uniqueness of this decomposition. 2
We are now ready to introduce our new interaction index, namely the mapping
IM : FN × 2N → R : (v, S) 7→ IM(v, S) = as(S),
where as(S) are the leading coefficients in (8) of the best sth approximation of ˆf , for all s = 0, . . . , n.
This immediately leads to the following equivalent definition, which makes use of the classical beta function
B(a, b) :=
Z 1
0 u
a−1(1 − u)b−1du (a, b > 0).
Definition 16 For any N = {1, . . . , n}, we define the mapping IM : FN × 2N → R as IM(v, S) = X T ⊇S qsta(T ), where qs t := (2s+1 s+1) (s+t+1 s+1 ) = B(s+1,s+1)B(t+1,s+1) for all t > s > 0.
By using Lemma 15, we can easily show that IM is a cardinal-probabilistic interaction index.
Proposition 17 The function IM : FN × 2N → R is a cardinal-probabilistic
interaction index, given by
IM(v, S) =
X
T ⊆N \S
ps
where ps
t(n) := B(n−t+1,s+t+1)B(s+1,s+1) .
Proof. We immediately observe that Lemma 15 applies with the beta distri-bution Fs(x) = Rx 0 us(1 − u)sdu B(s + 1, s + 1) . Indeed, we have Z 1 0 x t−sdF s(x) = 1 B(s + 1, s + 1) Z 1 0 x t(1 − x)sdx = qs t.
The expression of the coefficients ps
t(n) then follows immediately from (11). 2
The following proposition shows that the mapping v 7→ (IM(v, S))S⊆N is in-vertible and yields the transformation formula from (IM(v, S))S⊆N to a = (a(S))S⊆N.
Proposition 18 For any S ⊆ N, we have
a(S) = X T ⊇S hs tIM(v, T ). with hs t = (−1)t−s ³ s+t t ´ ³ 2t t ´ .
Proof. We only need to show that
X T ⊇S qs t X R⊇T ht r IM(v, R) = IM(v, S). We have X T ⊇S qs t X R⊇T ht r IM(v, R) = X R⊇S IM(v, R) X T : S⊆T ⊆R qs t htr,
where the inner sum equals 1 if S = R and 0 if S R. Indeed, in the latter case, we have
X T : S⊆T ⊆R qsthtr= ³ 2s+1 s+1 ´ ³ 2r r ´ (s + 1)! r! r X t=s (−1)r−t à r − s t − s ! (r + t)! (s + t + 1)! = ³ 2s+1 s+1 ´ ³ 2r r ´ (s + 1)! r! r X t=s (−1)r−t à r − s t − s !· dr−s−1 dxr−s−1 x r+t ¸ x=1 = ³ 2s+1 s+1 ´ ³ 2r r ´ (s + 1)! r! · dr−s−1 dxr−s−1x r+s(x − 1)r−s ¸ x=1 = 0. 2
In order to conclude, we focus on the power index associated to IM. The following corollary shows that, incidentally, this power index identifies with the index Wi( ˆf ) previously introduced axiomatically by Grabisch and Labreuche [10, Theorem 2] in the context of multicriteria decision making.
Corollary 19 The restriction of IM(v, ·) to singletons is given by:
IM(v, {i}) =
X
T ⊆N \{i}
6(n − t)!(t + 1)!
(n + 2)! [v(T ∪ {i}) − v(T )].
Appendix A: Proof of Lemma 6
Observe first that we can assume that S and T are such that |S ∪ T | = n. Moreover, suppose that S and T are nonempty. Remark that [0, 1]n is equal almost everywhere to the disjoint union of the sets
Sσ◦ := {x ∈ [0, 1]n| xσ(1) < · · · < xσ(n)} (σ ∈ Sn). Then we have Z [0,1]nminS(x) minT(x) dx = X σ∈Sn Iσ, (13) where Iσ := Z S◦ σ minS(x) minT(x) dx (σ ∈ Sn), Now, define for every R ⊆ N and p ∈ {1, . . . , |R|} the set
S(p)
n (R) := {σ ∈ Sn| σ(1), . . . , σ(p) ∈ R, σ(p + 1) 6∈ R}. It is then clear that there is a decomposition of Sn into disjoint subsets
Sn = · Sn(S ∩ T ) ¸ ∪ ·n−t[ p=1 S(p) n (S \ T ) ¸ ∪ ·n−s[ p=1 S(p) n (T \ S) ¸ , (14)
where Sn(S ∩ T ) := {σ ∈ Sn| σ(1) ∈ S ∩ T }. Moreover we have |Sn(S ∩ T )| = |S ∩ T | (n − 1)! |S(p) n (S \ T )| = ³ n−t p ´ p! t (n − p − 1)! .
Indeed, every element of Sn(S ∩ T ) corresponds to a choice of σ(1) in S ∩ T , and a permutation of the remaining elements in N. In the same fashion, every element of S(p)
n (S \ T ) is determined by the choice of p elements in S \ T , a permutation of these elements, the choice of a single element in T and a permutation of the remaining elements. Moreover, for any σ ∈ Sn(S ∩ T ), we have Iσ = Z 1 0 Z x σ(n) 0 · · · Z x σ(2) 0 x 2 σ(1)dxσ(1)· · · dxσ(n) = 2 (n + 2)!, and X σ∈Sn(S∩T ) Iσ = 2 |S ∩ T | (n − 1)! (n + 2)! = 2 (s + t − n) n(n + 1)(n + 2). (15)
In the same way, for any σ ∈ S(p)
n (S \ T ), we have Iσ= Z 1 0 Z x σ(n) 0 · · · Z x σ(p+2) 0 xσ(p+1) Z x σ(p+1) 0 · · · Z x σ(2) 0 xσ(1)dxσ(1)· · · dxσ(n) = p + 2 (n + 2)!, and n−tX p=1 X σ∈S(p)n (S\T ) Iσ= t (n + 2)! n−t X p=1 à n − t p ! p! (n − p − 1)! (p + 2) =t! (n − t)! (n + 2)! n−t X p=1 à n − p − 1 t − 1 ! (p + 2) =t! (n − t)! (n + 2)! " (n + 2) n−tX p=1 à n − p − 1 t − 1 ! − t n−t X p=1 à n − p t !# =t! (n − t)! (n + 2)! " (n + 2) à n − 1 t ! − t à n t + 1 !# , that is, n−t X p=1 X σ∈S(p)n (S\T ) Iσ = (n − t)(n + 2t + 2) n(n + 1)(n + 2)(t + 1). (16)
Similarly, we can write n−sX p=1 X σ∈S(p)n (T \S) Iσ = (n − s)(n + 2s + 2) n(n + 1)(n + 2)(s + 1) (17)
and by Eqs. (13)–(17), we have
Z [0,1]nminS(x) minT(x) dx = X σ∈Sn(S∩T ) Iσ + n−t X p=1 X σ∈S(p)n (S\T ) Iσ+ n−sX p=1 X σ∈S(p)n (T \S) Iσ = s + t + 2 (n + 2)(s + 1)(t + 1)
as desired. We can easily see that the result still holds if S = ∅ or T = ∅. 2
Appendix B: Proof of Lemma 7
We only have to prove that the explicit expression we give for Ak(minS) fulfills the conditions of Proposition 3, namely,
D
minS− Ak(minS), minT
E = 0 (T ⊆ N, t 6 k), or, equivalently, X J⊆S IJ,T(−1)k+j à k + j + 1 k + 1 ! = 0 (T ⊆ N, t 6 k), (18) where IJ,T := hminJ, minTi is given explicitly in (6).
Partitioning J ⊆ S into P ⊆ S \ T and Q ⊆ R with R := S ∩ T (hence
r 6 t 6 k), the left-hand side of (18) becomes
X P ⊆S\T X Q⊆R IP ∪Q,T (−1)k+p+q à k + p + q + 1 k + 1 ! = k−r+1X p=0 à k − r + 1 p ! r X q=0 à r q ! p + q + t + 2 (p + t + 2)(p + q + 1)(t + 1)(−1) k+p+q à k + p + q + 1 k + 1 ! = − k−r+1X p=0 à k − r + 1 p ! (−1)k−r+1−p 1 p + t + 2 r X q=0 à r q ! (−1)r−qg(p + q) where g : N → R is defined by
g(z) = z + t + 2 (z + 1)(t + 1) Ã z + k + 1 k + 1 ! = 1 k + 1 Ã z + k + 1 k ! + 1 t + 1 Ã z + k + 1 k + 1 ! .
Consider the difference operator
∆nf (n) := f (n + 1) − f (n) for functions on N. It is well-known that we have
∆k nf (n) = k X j=0 Ã k j ! (−1)k−jf (n + j) (k ∈ N). (19)
Applying this to g, we obtain r X q=0 Ã r q ! (−1)r−qg(p + q) = ∆r pg(p) = 1 k + 1 Ã p + k + 1 k − r ! + 1 t + 1 Ã p + k + 1 k − r + 1 !
and the left-hand side of (18) becomes
− k−r+1X p=0 Ã k − r + 1 p ! (−1)k−r+1−p 1 p + t + 2 µ 1 k + 1 Ã p + k + 1 k − r ! + 1 t + 1 Ã p + k + 1 k − r + 1 !¶ .
Applying (19) again, we see that this latter expression can be written as
− ∆k−r+1 z 1 z + t + 2 µ 1 k + 1 Ã z + k + 1 k − r ! + 1 t + 1 Ã z + k + 1 k − r + 1 !¶ z=0 .
We now show that the expression in brackets is identically zero. We do this by considering two cases:
• If t = k then 1 z + t + 2 µ 1 k + 1 Ã z + k + 1 k − r ! + 1 t + 1 Ã z + k + 1 k − r + 1 !¶ = 1 (k + 1)(z + k + 2) Ã z + k + 2 k − r + 1 ! = 1 (k + 1)(k − r + 1) Ã z + k + 1 k − r !
is a polynomial Pk−r(z) of degree k − r, and ∆k−r+1z Pk−r(z) ≡ 0.
• If r 6 t 6 k − 1 then 1 z + t + 2 µ 1 k + 1 Ã z + k + 1 k − r ! + 1 t + 1 Ã z + k + 1 k − r + 1 !¶ = 1 z + t + 2 µ 1 (k + 1)(k − r)! k+1Y i=r+2 (z + i) + 1 (t + 1)(k − r + 1)! k+1Y i=r+1 (z + i) ¶
is a polynomial Qk−r(z) of degree k − r, and ∆k−r+1z Qk−r(z) ≡ 0. 2
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